Simulation Measurement Error

Measurement Error

Suppose \[\begin{align*} Y_i = & \beta_0 + \beta_1 X_i^* + e_i^*,\\ X_i = & X_i^* + v_i. \end{align*}\]
where \(X_i\) is observed variable and \(X_i^*\) is true value. Suppose \[\mathbb{E}[e_i^*] =\mathbb{E}[e_i^* X_i^*] = 0,\] \[\mathbb{E}[v_i] = \mathbb{E}[X_i^* v_i] = \mathbb{E}[e_i^* v_i] = 0.\] Let \(\sigma^2_\nu = \mbox{Var}(\nu_i)\). Consider regressing \(Y_i\) on observed \(X_i\).

Simulation

Consider regressing \(Y_i\) on observed \(X_i\) in simulation with following DGP:

\[\begin{eqnarray*} \beta_0 & =0, ~~ \beta_1 =1, \\ X_i^* & \sim N(0,1),\\ \epsilon_i^* & \sim N(0,0.16),\\ \nu_i & \sim N(0,\sigma^2_\nu) \end{eqnarray*}\] with \(\sigma^2_\nu = 0, 0.25, 1,\) or \(4\), and with \(N=4,000\).